The Mathematics Survey:
A Tool for Assessing
Attitudes and Dispositions
T
he need for positive attitudes and dispositions permeates the teaching and learning of
mathematics. What students believe about
mathematics influences what they are willing to
say publicly, what questions they are likely to
pose, what risks they are willing to take, and what
connections they make to their lives outside the
classroom (Borasi 1990; Whitin and Whitin 2000).
Unless students have a realistic sense of mathematical applications in real-life contexts, they
are unlikely to see themselves pursuing courses in
advanced mathematics or choosing mathematicsrelated careers (Picker and Berry 2001).
Principles and Standards for School Mathematics (NCTM 2000) delineates a range of attitudes
and beliefs about mathematics that contribute to
productive problem solving and communication.
For example, perseverance, curiosity, confidence,
and flexible thinking are related to learners’ investment in challenging problem solving and investigations involving complex patterns and relationships.
Confidence, open-mindedness, a willingness to
share one’s own successes and failures, and the
ability to shift perspectives are hallmarks of meaningful communication. Resourcefulness and reflective analysis are important dimensions in learners’
ability to use various forms of representation to
generate, clarify, and express thinking.
Given the ways in which mathematical attitudes,
skills, knowledge, and strategies are intertwined,
assessing students’ attitudes and beliefs can proBy Phyllis E. Whitin
Phyllis E. Whitin, phyllis.whitin@wayne.edu, is associate professor of elementary education at Wayne State University, Detroit, MI 48202. Her research
interests include the integration of language arts, mathematics, and science
as well as inquiry-based learning.
426
vide valuable information, especially at the beginning of the school year. The results can be used to
guide the development of a classroom environment
conducive to growth in positive attitudes and in
addressing misconceptions and counterproductive
beliefs (Picker and Berry 2001; Rock and Shaw
2000). This article describes the development and
implementation of a mathematics attitude survey
designed to meet this need. Examples of fourthgrade children’s responses to the survey over a
four-year period, the ways in which the results
guided teaching and learning, and an analysis of
post-assessments illustrate the process.
The Mathematics Survey as
a Tool for Assessment
The mathematics survey (fig. 1) is composed of
six sentence-completion prompts designed to
elicit children’s perceptions of what constitutes
mathematical knowledge, ways of thinking, and
usefulness in everyday life. It was adapted from
the Burke Reading Survey (Goodman, Watson, and
Burke 1987) and correlates with attitudes identified
by NCTM (2000). Prompts 1, 2, 3, and 5—“To be
good in math, you need to ... because ...”; “Math is
hard when ...”; “Math is easy when ...”; “The best
thing about math is ...”—are included to encourage responses that reflect the degree of students’
confidence, curiosity, flexible thinking, and their
views of student-teacher and student-student relationships. The fourth prompt—“How can math
help you?”—addresses students’ perceptions about
the usefulness of mathematics and real-life applications. The final prompt—“If you have trouble solving a problem in math, what do you do?”—is intentionally somewhat ambiguous so that the students
can reveal their definitions of, attitudes toward, and
strategies for problem solving.
Teaching Children Mathematics / April 2007
Copyright © 2007 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.
This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
The following questions guided the analysis of
student responses:
• What do these responses reveal about students’
perceptions regarding the teacher-student relationship? Regarding student-student relationships? Do references to the teacher imply that
the teacher is the sole source of knowledge? Is
there evidence of student autonomy? Of collaborative thinking? Are other students mentioned?
If so, in what ways?
• What do these responses reveal about students’
perceptions regarding mathematical content and
applications? Do students cite examples of functional applications of mathematical ideas? Do
they view mathematics as valuable and relevant
in both the present and the future?
• What do these responses reveal about students’
perceptions regarding processes of engaging
in mathematical investigations (e.g., planning,
reasoning, using strategies in a flexible manner,
making connections, representing ideas in multiple ways, collaborating, discovering patterns
and relationships)? Do the students view challenge as rewarding?
Examples of typical responses collected from the
four fall surveys illustrate the assessment process
(see fig. 2). In these examples, words and phrases
about listening, paying attention, and studying
Teaching Children Mathematics / April 2007
Figure 1
Mathematics survey
1.
2.
3.
4.
5.
6.
To be good in math, you need to ... because ...
Math is hard when ...
Math is easy when …
How can math help you?
The best thing about math is ...
If you have trouble solving a problem in math, what do you do?
Tell anything else you want about math.
On the back of your paper, draw a picture that shows what math means to
you.
suggest a belief that mathematics is a silent and
solitary endeavor. The responses suggest that these
children view mathematics class as a period of
teacher-student interchanges in which the teacher
poses questions or problems and evaluates answers.
The children do not usually mention their peers
except in a negative sense, such as “talking to other
people” rather than “paying attention.” To be a successful mathematics student, one must listen to the
teacher, follow directions, and study. Responses
such as this to prompt 1—“study real hard because
you need to know the problems in a flash”—also
imply that speed is a universal measure of mathematical success. References to extended investigations are absent. Similarly, each year about half the
responses to prompt 6—“If you have trouble solving a problem in math, what do you do?”—suggest
427
Figure 2
Typical student responses from the fall mathematics surveys
1. To be good in math, you need to ... because …
“study and do all your papers”
“concentrate and don’t play around because if you don’t concentrate you
will do bad”
“study and listen because you might not know what to do”
“study real hard because you need to know the problems in a flash”
2. Math is hard when ...
“it’s 2-digit division”
“you don’t listen, not pay attention, or talking to other people”
“you don’t concentrate”
“you don’t follow directions”
“you can’t understand what the teacher says”
3. Math is easy when ...
“you work good by listening to the teacher”
“when you have studied really hard”
“when it’s times or plus”
4. How can math help you?
“when you get good grades and not get on restriction”
“when you go to the store”
“when you get a job”
“when you get an A”
“If you saw two numbers and if you wanted to do something with the
numbers­.” …
6. If you have trouble solving a problem in math, what do you do?
“Raise your hand and ask the teacher.”
“Ask for help.”
“I skip it and do the next one.”
that students depend on the teacher for direction
(e.g., “raise your hand,” “ask the teacher”). Other
responses such as the one suggesting skipping the
problem imply time management. Only rarely do
students describe devising an alternative method,
collaborating, or using various forms of representation as problem-solving strategies.
In response to prompt 4—“How can math help
you?”—the children’s statements indicate that
the rewards of engaging in mathematical activity
are extrinsic—for example, getting good grades
and maintaining good relationships with parents.
Almost exclusively, students do not mention the
intrigue or challenge of investigations as rewarding
or “fun.” Few students mention mathematics as useful in the present; most give vague references to college or employment in the distant future. Although
some children cite using money as a helpful part of
mathematics, others mention computational activity that is devoid of any context, such as the student
who suggested seeing “two numbers” and “wanting
to do something” with them.
The patterns that emerged from the survey analysis illuminate a range of attitudes and beliefs that
could interfere with students’ mathematical growth.
428
Identifying these trends served to guide plans for
structuring the environment, designing instructional activities, and delineating the teacher’s role.
Figure 3 shows a summary of the trends from the
fall surveys; productive attitudes, dispositions,
and beliefs about mathematics (NCTM 2000); and
plans for teaching and learning.
Using the Results of the
Survey
Changing the view of mathematics as a solitary
endeavor entailed structuring group problem-solving tasks, promoting collaborative conversations,
and encouraging the children to view their peers
as resources. In the case of these fourth graders,
the teacher decided that beginning the year with
a noncomputational activity, such as pentominoes
or classification with attribute blocks, could help
expand the children’s limited views of mathematics
as computation. Further, a puzzle or game format
could highlight for the children that all mathematical activity is not done “in a flash” or by following
a prescribed procedure. As pairs or small groups
of children worked together to solve puzzles, their
conversations laid an important foundation for writing and visual representation (Huinker and Laughlin 1996; Whitin and Whitin 2003; Wickett 1997).
At the completion of the task, the teacher conducted
a reflective conversation that specifically addressed
the targeted dispositions. She posed questions such
as these: “What was going through your mind
when you first started the puzzle?” “How did your
group’s ideas help you?” “What did you do if your
first idea didn’t work?” Many children were surprised to find that their peers encountered frustration or that the teacher did not regard their building
off a classmate’s suggestion as “cheating.”
Following the conversation, the teacher asked
the children to record their discoveries about their
learning processes in writing and to name specific children whose thinking helped them—for
example, “Catherine helped me when she said,
‘Switch them around.’” Public acknowledgment of
collaborative efforts and written reflections about
problem-solving processes continued throughout
the year (Whitin and Whitin 2000). Figure 4 shows
one student’s representation of the value of mathematical conversations. The first box shows a red
and a blue circle, representing two children’s ideas.
During the conversation, the two ideas begin to
blend (second box) until finally they merge into one
purple circle in which “the class works together and
Teaching Children Mathematics / April 2007
the ideas get mixed.” The student’s summary—“I
think math is easier when our class puts our ideas
together”—demonstrates a marked change in attitude from such presurvey comments as “Math is
easy when it’s times or plus.”
To build the students’ confidence and selfreliance, the teacher needed to shift attention away
from herself as dispenser of knowledge. To achieve
this goal, she carefully examined the implicit messages conveyed through her interactions with the
students. She made conscious efforts to respond to
the children’s questions and comments in ways that
invited revisiting or extending a problem (Schwartz
1996). If the students asked, “Is that right?” the
teacher, to encourage them to revisit their thinking
process and either confirm or revise their solution,
would respond, “How can you be sure?” or “Explain
your thinking.” When the children shared a conjecture—for example, “When you add two odd numbers, you get an even number”—the teacher invited
further investigation by asking, “Does that always
work?” If a child raised a question during a class
discussion, the teacher developed the habit of turning the question over to the group. In addition, she
learned to carefully choose words that conveyed
resourcefulness and curiosity—for example, invent
or discover rather than find or use.
Regularly using student-authored problems for
homework and a morning “problem of the day”
were additional ways to increase the children’s
confidence. Over time, the teacher strove to feature
as the “problem of the day” an original problem
written by every child in the class. As part of the
ritual, the student-author led the discussion of the
problem’s solution (Whitin and Whitin 2000). On
other occasions, the children expanded on entries
in their mathematics journals and “published” their
findings for parents and other classes. In these ways
the children had opportunities throughout the year
to view one another as resources, develop their
confidence, and feel recognized and rewarded for
their learning.
Instituting a mathematical forum where students
shared their problem-solving strategies served to
develop the children’s flexible thinking. When one
child demonstrated his use of the distributive property, for example, the teacher suggested that the
other students work in groups to apply his strategy
to a variety of problems and later share their discoveries. The teacher also introduced the students to
problem posing (Brown and Walter 1990). Sometimes she planned problem-posing explorations in
advance (Whitin forthcoming), and on other occaTeaching Children Mathematics / April 2007
sions she built on children’s spontaneous questions,
such as, “If 4 × 3 = 3 × 4, why doesn’t 34 × 3 = 33
× 4?” This question inspired an investigation of the
commutative property in multiplication, as well as
the problem-posing extension, “Are there similar
multiplication problems that do yield the same
product? What are the attributes of the problems
that do ‘work’?” In pursuing these extensions, one
child found that 22 × 3 equaled 33 × 2. Following the pattern of using a two-digit number with
the same number of tens and ones, she tried 33 ×
6 and 66 × 3. When she discovered that the strategy worked, she wrote: “When I wrote down the
problems I didn’t know if I would get the same
answer or not. I used these numbers because 3 × 6 =
18 and 6 × 3 = 18.” Her statement conveys her
willingness to take risks, her resourceful thinking,
and her sense of personal accomplishment. These
Figure 3
Using the analysis of survey results to guide instructional plans
Trends Implied by
the Surveys
Positive Attitudes,
Dispositions, Beliefs
Instructional Plans
for Change
1. Mathematics is
a solitary, silent
endeavor.
1. Collaboration
and communication contribute
to mathematical
understanding.
1. Structure group tasks;
make children’s strategies public; encourage
children to note others’
contributions to their
learning
2. The teacher is in
charge of imparting knowledge.
The rewards for
developing mathematical expertise
are external and are
often postponed
until the future.
2. Mathematics involves learners in
constructing meaning for themselves.
The rewards for
developing expertise are intrinsic.
2. Encourage interaction,
revisiting, extending
(Schwartz 1996); involve
students through
­student-authored­
problems, mathematics
journals, mathematics
“publications”
3. Problems are
solved in a swift,
prescribed manner.
3. Problems are
solved through
flexible use of
multiple strategies.
The time required
to solve problems
depends on the
complexity of the
problem.
3. Encourage strategy
sharing, problem-posing
investigations, extended
explorations, mathematics journals (Whitin and
Whitin 2000)
4. Mathematics is
unrelated to other
subjects.
4. Mathematics has
real-life application across the
curriculum and in
contexts outside
school.
4. Emphasize contentrelated­ problems (e.g.,
science), problems
inspired by children’s
literature, studentauthored­ problems
429
Figure 4
A student’s visual depiction of the value of sharing ideas in
mathematical problem solving
and other problem-posing explorations also helped
dispel the notion that all mathematical proficiency
is universally equated with the speed of generating
a solution.
Finally, problem-solving opportunities arose in
contexts outside mathematics class. Students gathered, represented, and analyzed data to make decisions about the ideal seed mixture to place in the
class bird feeders (Whitin and Whitin 1999). While
studying geology in science, the children used nets
to build models of crystals, a process that afforded
them the opportunity to apply geometric principles
and terminology to the natural world (see fig. 5).
In addition, the teacher regularly read aloud mathematics-related children’s literature to demonstrate
mathematical connections within a wide variety
of contexts, initiate investigations, and inspire the
children’s writing. The changes in children’s views
about the usefulness of mathematics, as well as
other attitudes and dispositions, were later reflected
in their end-of-the-year assessment.
The Survey as PostAssessment: Reflection
and Evaluation
At the end of each year, the children completed
the survey as a post-assessment. Usually the same
survey format was used for both the pre-assessment­
and the post-assessment, but in the final year the
post-assessment survey was slightly modified.
Question 4—“How can math help you?”—was
430
changed to “To think mathematically means ...”
so that the children would state more directly their
definitions of mathematical activity. Two new
prompts were added: “When you write and draw
about math ...” and “ ‘What if’ in math. ....” The first
would ensure that the children address multiple
forms of representation and communication, while
the second referred to problem-posing experiences
(Brown and Walter 1990). Both questions were
included as a means to evaluate the effectiveness of
the instructional modifications made to address the
needs identified on the fall surveys. The examples
in figure 6 illustrate trends in the end-of-the-year
assessments.
These examples of responses show a wider
variety than those from the fall surveys. This range
suggests that over the course of the year, the children developed more individualized mathematical
identities as well as the confidence to express themselves. One of the sharpest contrasts with the initial
surveys is that the later surveys contain almost no
references to the teacher. For a student facing a
difficult problem, asking “someone to help me”
implies peers as well as adults. This student also
shows responsibility by adding, “I’ll do the rest.”
In this case, “help” does not mean that “someone
else will do the work for me.” The student who
mentioned the teacher directly included other alternatives as well. Responses such as “use your own
or someone else’s strategy” and “Math is easy when
there are groups or partners” also show an appreciation for collaborative thinking. In fact, as a whole,
the children used the pronoun “we” in phrasing
their answers to various prompts. This subtle shift
in language from the fall surveys further implies a
collaborative spirit.
In addition to showing less dependence on the
teacher, the students showed more resourcefulness in their attitudes about problem solving. The
response “make it into an easy math problem” suggests the strategy of simplifying the problem, while
“trying another problem to help solve that problem”
and “relate other math to it” show students’ awareness of connections among mathematical ideas.
The child who noted “I write down what I think”
is aware that writing is a tool for reflection and
discovery.
Sample responses to two of the revised and
new questions are shown in figure 7. The collection of statements implies active construction of
mathematical meaning. Thinking mathematically
incorporates strategic thinking (“math strategies”),
the ability to assume multiple perspectives or pose
Teaching Children Mathematics / April 2007
Figure 5
Photographs by David J. Whitin; all rights reserved
While studying geology in science, the children used nets to build models of crystals, a process that afforded them the
opportunity to apply geometric principles and terminology to the natural world.
5a. Building a model of a calcite crystal
5b. Comparing a calcite crystal with its model
problems (“say a lot of things about one question”),
ownership (“make up creative problems”), and
responsibility for one’s own learning (“be energetic
and serious about your attitude and thinking”).
Some responses to the new question “ ‘What if’
in math ...” revealed an appreciation for engaging in
mathematical investigations—for example, “everything would be more of a challenge and a mystery,
math would be even more fun.” Interestingly, the
child who initially responded to the prompt “How
can math help you?” with the abstract example of
“wanting to do something with 2 numbers” commented in the spring that “you can use math with
almost everything.” In contrast to earlier comments
about “getting an A” and benefiting from mathematics “in college,” these comments convey a sense
of mathematical activity as intrinsically rewarding. Thus, analysis of the post-assessment surveys
provided documentation of individual children’s
growth as well as trends in the classroom community. This feedback was valuable in the ongoing
process of refining teaching throughout the four
Teaching Children Mathematics / April 2007
years of this study.
Children’s attitudes and dispositions play a
vital role in mathematics classrooms. The survey
described here suggests one way to gain a window
into children’s existing beliefs. Given that information, teachers can better make instructional plans to
help their students become more confident, enthusiastic, and autonomous learners.
References
Borasi, Raffaella. “The Invisible Hand Operating in
Mathematics Instruction: Students’ Conceptions and
Expectations.” In Teaching and Learning Mathematics in the 1990s, 1990 Yearbook of the National Council of Teachers of Mathematics (NCTM), edited by
Thomas J. Cooney, pp. 174–82. Reston, VA: NCTM,
1990.
Brown, Stephen, and Marion Walter. The Art of Problem
Posing. Hillsdale, NJ: Lawrence Erlbaum, 1990.
Goodman, Yetta, Dorothy J. Watson, and Carolyn L.
Burke. Reading Miscue Inventory: Alternative Procedures. New York: Richard C. Owen, 1987.
Huinker, DeAnn, and Connie Laughlin. “Talk Your Way
431
Figure 6
Typical student responses from post-assessment mathematics ­surveys
1. To be good in math, you need to ... because …
“try hard and correct yourself because you would have a hard time if you don’t”
“have your own or someone else’s strategy”
“be able to solve puzzles and be able to make puzzles because it means you know what math is”
“study on what you do good because then you will do a lot better”
2. Math is hard when …
“you don’t have a strategy and you can’t find a strategy”
3. Math is easy when …
“you really understand it”
“you have had experience”
“there are groups or partners”
…
6. If you have trouble solving a problem, what do you do?
“If it is times I do plus. If it is division I do times.”
“I try my hardest and write down what I think.”
“I relate other math to it and work from there.”
“I ask someone to help me a little and I’ll do the rest.”
“Think slow so you will not be confused and you might need help from your teacher or use paper.”
“You can look at a hard problem and make it into an easy math problem.”
Figure 7
Typical student responses to sample
revised and new survey questions
When you write and draw about math ...
“if you draw a picture, then you will find the
answer”
“you can understand it better”
“it puts new thoughts in your head”
“you are showing how you think in math”
“I have a feeling I will make lots of
­connections”
To think mathematically means …
“to think hard about your math and make up
creative problems”
“you use math strategies to solve a problem”
“you can say a lot of things about one
question­”
“to be energetic and serious about your attitude and thinking”
into Writing.” In Communication in Mathematics
K–12 and Beyond, 1996 Yearbook of the National
Council of Teachers of Mathematics (NCTM), edited
by Portia C. Elliott, pp. 81–88. Reston, VA: NCTM,
1996.
National Council of Teachers of Mathematics (NCTM).
Principles and Standards for School Mathematics.
Reston, VA: NCTM, 2000.
Picker, Susan W., and John S. Berry. “Your Students’
Images of Mathematicians and Mathematics.” Mathematics Teaching in the Middle School 7 (December
2001): 202–8.
Rock, David, and Jean Shaw. “Exploring Children’s
432
Thinking about Mathematicians and Their Work.”
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55.
Schwartz, Sydney L. “Hidden Messages in Teacher Talk:
Praise and Empowerment.” Teaching Children Mathematics 44 (March 1996): 396–401.
Whitin, David J., and Phyllis Whitin. “Building a Community of Mathematicians.” Teaching Children Mathematics 9 (April 2003): 464–69.
Whitin, Phyllis. “Meeting the Challenges of Negotiated
Mathematical Inquiry.” Teaching and Learning: The
Journal of Natural Inquiry and Reflective Practice
(2006–7) (forthcoming). www.und.nodak.edu/dept/
ehd/journal/.
Whitin, Phyllis, and David J. Whitin. “Mathematics Is
for the Birds: Reasoning for a Reason.” In Developing Mathematical Reasoning, K–12, 1999 Yearbook
of the National Council of Teachers of Mathematics
(NCTM), edited by Lee Stiff, pp. 107–14. Reston,
VA: NCTM, 1999.
———. Math Is Language Too: Talking and Writing in
the Mathematics Classroom. Urbana, IL: National
Council of Teachers of English and Reston, VA: National Council of Teachers of Mathematics, 2000.
Wickett, Maryann. “Uncovering Bias in the Classroom—
A Personal Journey.” In Multicultural and Gender
Equity in the Mathematics Classroom: The Gift of
Diversity, 1997 Yearbook of the National Council of
Teachers of Mathematics (NCTM), edited by Janet
Trentacosta and Margaret J. Kenney, pp. 102–6. Reston, VA: NCTM, 1997. s
Teaching Children Mathematics / April 2007
Reflect and Discuss:
The Mathematics Survey: A Tool for Assessing Attitudes and
Dispositions
Reflective teaching is a process of self-observation and self-evaluation. It means looking at your classroom practice, thinking about what you do and why you do it, and then evaluating whether it works. By collecting information about what goes on in our classrooms and then analyzing and evaluating this information, we identify and
explore our own practices and underlying beliefs.
The following questions related to “The Mathematics Survey: A Tool for Assessing Attitudes and Dispositions” by Phyllis Whitin are suggested prompts to aid you in reflecting on the article and on how the author’s
ideas might benefit your own classroom practice. You are encouraged to reflect on the article independently as
well as discuss it with your colleagues.
• What is the relationship between attitudes and dispositions and mathematical proficiency?
• How do your students’ comments and actions in the classroom suggest their attitudes and dispositions?
Share your observations and analysis with your colleagues.
• What strategies do you use to foster students’ development of positive attitudes and dispositions­?
• Children’s views of geometry or data analysis are sometimes very different from their views of number
or algebra. What questions would you use to assess students’ attitudes and dispositions toward the next
mathematics unit you are scheduled to teach? How might the data change how you go about teaching
the unit?
• Teachers, particularly novice teachers, tend to rely on lesson planning formats to consider the important elements of the lesson. How might one modify the lesson plan format to remind teachers that attention to issues of attitudes, confidence, and collaboration are important?
You are invited to tell us how you used “Reflect and Discuss” as part of your professional development. The
Editorial Panel appreciates the interest and values the views of those who take the time to send us their comments. Letters may be submitted to Teaching Children Mathematics at tcm@nctm.org. Please include “Readers’
Exchange” in the subject line. Because of space limitations, letters and rejoinders from authors beyond the 250word limit may be subject to abridgement. Letters also are edited for style and content.
Hot Topics
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meet the needs of both prospective and “seasoned” teachers:
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